They are not in CrossFire.
Well... then I'll just prattle off theories that sound plausible:
*Power issue. Could be a bad molex cable/adapter, or something wrong with the PSU itself (though this seems unlikely since there's nothing wrong with the other). I'm guessing you're not in an area with frequent brownouts... I'd switch the molex cables on the 6870 with the ones on the 6970, reboot, and see if issue is solved. (or if PSU is modular, just use different molex cables for 6970)
*Your OS or BIOS is trying to switch which GPU (between integrated/dedicated) it's using for display, causing some type of software/driver problem. Obviously not the problem if you don't have an motherboard-integrated GPU.
Other ideas:
-Switch which GPU you connect a monitor to, or try a dummy plug in the 6970.
-Switch out PSU with known-good, if available. Re-test.
-Switch the position of the 6970 and 6870 on the motherboard. It's possible there's an issue with the PCI-E port on the mobo.
-If you have multi-boot PC, try a different OS and re-test. If you're on Windows and maybe don't have much Linux experience, Ubuntu's pretty easy to install with LiveCD versions.
-Take out the 6870 and see if the 6970 still drops load. If not, and other issues are proven non-issues, there may be something wrong with the motherboard itself.
-If all else fails and nobody comes in with more ideas.... try running Furmark or a similar gfx benchmark tool and see if the load drops on there, too.
